What a missing end does to the weave

This collection has one central question and asks it of every draft it draws: is this one cloth? A matrix of filled and empty squares can describe fabric that falls into two independent layers, or that has threads lying loose on the surface, and nothing in the drawing betrays either.

The question has never been asked of a damaged cloth. Every draft this collection has examined has been the draft somebody intended. So here is the natural next question, and it has an exact answer: take every four-by-four draft in which each end and each pick interlaces — 22,874 of them — remove each of its four ends in turn, and ask the criterion what it makes of what is left.

Ninety-one thousand four hundred and ninety-six trials. The answer is not what the question suggests.

The claim #

A missing end never splits a four-by-four cloth into layers. What it does instead, in 44.5 per cent of cases, is leave a pick with nothing holding it — which can be pulled out of the fabric with the fingers.

The zero is exact and it is not an accident of the census: it follows from what deleting a column does. And the 44.5 per cent is not a small number: nearly half of all the ways a four-by-four cloth can lose an end produce a thread that is not woven in at all.

Only 834 of the 22,874 drafts survive losing any of their four ends with every remaining thread still interlaced. Twenty-two thousand and forty of them have at least one end whose loss leaves something loose.

The argument for the zero #

A separation is a statement about a relation being pointed the wrong way. The criterion this collection uses reads every crossing as this thread is above that one, builds the digraph of those relations, and asks whether it is strongly connected: the cloth hangs together exactly when every thread can be reached from every other by going up.

Deleting an end deletes relations. It does not reverse any. Reachability is the whole of the criterion, and a deleted relation cannot create a set of threads that is above everything it crosses. Every remaining crossing points where it pointed before, so every path that existed between two remaining threads either survives intact or loses one of its steps — and at this repeat, the surviving paths are enough. A cloth cannot be split by taking constraints away when the constraints that remain are all still pointing the same way.

Reversing one, on the other hand, can split it. That is the next essay’s subject and it is worth stating here because the two together make an inversion: one wrong intersection can break a cloth into layers and a whole missing thread cannot. The smaller fault is the more dangerous one.

Why the zero is a property of the repeat #

A count that comes back zero is the one that most needs a control, because zero is also what a broken enumeration returns.

So the same operation was applied at a larger repeat, where there is more cloth to fall apart and where the answer is not known in advance: four thousand random six-by-six drafts, each filtered to be one cloth with everything interlaced to begin with, each losing every one of its six ends in turn.

Thirty-seven of the twenty-four thousand trials separate. So a missing end can split a cloth — it simply cannot at four by four, where the repeat is too small for the remaining relations to leave anything stranded.

That is the honest form of the result: a rule at one size and a rarity at the next. The zero is real, it is checked, and it is not a theorem about weaving.

The outcome the criterion has to report differently #

The interesting half of this census is not the zero. It is that the criterion, asked about a damaged cloth, returns an answer it was never designed to give.

A draft with a loose thread in it cannot be asked the integrity question at all. The criterion is about which threads hold which down, and a thread that holds nothing and is held by nothing makes the question ill-posed: it is neither part of the cloth nor a separate layer of it. It is a thread lying in a fabric.

So the outcomes have to be three rather than two — sound, loose, separated — and the loose case has to be reported as its own thing rather than counted as a separation. The first version of this census did count it as one, and put nine hundred drafts in the wrong column.

What happens to the float #

Of the fifty thousand sound outcomes, what has changed?

The picks that were held by the missing end are now held by whatever is on either side of it, so the float across the ends grows. Measuring it correctly needs care: the reduced draft is one end narrower, so counting runs in the reduced matrix reports floats getting shorter when the cloth’s floats are getting longer. Every run has to be measured in the width the cloth had, with the missing end’s place counted as part of whatever crosses it.

Done that way, the answer is uniform to the point of being funny. Every sound outcome has a longest float of exactly three end widths — for a plain weave that is a rise from one, for a twill from two or three, and for everything else no change at all, since most drafts already carry a three somewhere.

So the fault is most conspicuous in the cloth that has the least float to begin with. A plain weave’s longest float trebles; an eight-end satin’s does not move. That is the same ordering which weave hides a fault arrives at from the other direction, and the agreement between two different perturbations is worth more than either.

What was counted, and how #

Every draft in the standing census is taken, each of its four ends deleted in turn, and the result put through the same three questions the collection asks of an intended draft: is every thread still interlaced, is what remains one cloth, and what is its longest float.

Four things are asserted while it runs.

• The outcomes sum to the trials. Ninety-one thousand four hundred and ninety-six, which is the census times four. A classification that loses cases is the failure this catches.
• Some drafts leave a pick loose, so the loose outcome is reachable and the classifier is not simply never firing.
• No draft separates, asserted as a count of zero rather than assumed away — which is the finding, and would fail loudly if the criterion were being handed a reduced draft in the wrong shape.
• A float grows somewhere, which would fail if the span arithmetic had been done in the narrower repeat.

And the control at six by six is run as part of the same sweep, with its own assertion that separations do occur there, so the zero above cannot quietly become a bug in the criterion without something going red.

Which drafts survive, and what they have in common #

Eight hundred and thirty-four drafts of the 22,874 lose any of their four ends and leave every remaining thread still interlaced. That is 3.6 per cent, and the survivors are not a random selection.

A draft survives losing an end when every pick still changes side among the three ends that remain. A pick that was over two ends and under two is over two and under one afterwards — fine. A pick that was over three and under one loses its single binding end and is over three of three — loose. So what decides survival is how evenly each pick is distributed across the ends, and how many different ends do the binding.

That gives a rule with a familiar shape, and the census makes it exact: every one of the ninety balanced drafts survives losing any of its four ends, and only 744 of the 22,784 unbalanced ones — 3.3 per cent — manage it. A pick with two ends up and two down has two chances of keeping a binder; a pick with three up and one down has one, and loses it whenever that one is the end that broke.

The connection to this collection’s standing result is worth making explicit. Balance forces integrity — every balanced four-by-four draft describes one cloth, which was one of the first findings this census produced — and here balance turns out to protect against damage as well. The same property that guarantees the intended cloth hangs together is what makes it robust to losing a thread.

And it explains the plain weave’s peculiar position. A plain weave is maximally balanced and survives a broken end in the sense that nothing comes loose in a wider repeat — but its float trebles, so the fault is maximally visible. Robustness and inconspicuousness are different virtues and this is a case where a cloth has one and not the other.

Eight hundred and thirty-four is a closed count #

The survivors were described above by a rule and counted by enumeration. The rule can be sharpened until it determines the count exactly, which is worth doing because the two then check each other — and because the sharpened rule is not quite the one the census suggested.

A pick survives losing any single end when it still has both a face and a back among the three that remain, which means it must have had at least two of each to begin with. On a repeat of four, at least two ones and at least two zeros forces exactly two of each. So

a draft survives losing any of its ends exactly when every pick is balanced — two up, two down — with no condition on the ends at all beyond the interlacing the census already imposes.

That is a condition on rows only, and it is weaker than balance in this collection’s usual sense, which asks it of rows and columns together. The ninety fully balanced drafts are a subset of the survivors, which is why every one of them survives.

Counting them is inclusion–exclusion. Each row is one of the C(4,2) = 6 balanced patterns, so there are 6⁴ = 1,296 matrices with every pick balanced; from those, remove the ones with a constant column, which would fail the interlacing condition. A single column constant leaves 162 such matrices; two columns, 34; three, 6; four, 6. So

1,296 − 4·162 + 6·34 − 4·6 + 6 = 834.

Which is the census’s number, arrived at without the census. Two derivations of one count, one by enumerating twenty-two thousand drafts and one by counting patterns, and they agree exactly — which is the check that the classifier is putting drafts in the right column and that the rule is the whole rule rather than most of it.

Why the rule is so clean at four and will not be at six #

The step that made it clean is worth isolating, because it is the step that does not generalise.

“At least two ones and at least two zeros” is a genuine condition at every even repeat. At a repeat of four it collapses into “exactly two”, because four bits cannot have three of one and two of the other. At six it does not collapse: a pick may have two, three or four ends up and still survive, which is 15 + 20 + 15 = 50 of the 64 possible picks rather than 6 of 16.

So the survival rate rises steeply with the repeat. At four, only 1,296 of 65,536 matrices have every pick eligible — two per cent. At six the corresponding fraction is (50/64)⁶, which is twenty-two per cent, before the constant-column corrections.

A larger repeat is far more robust to a broken end, and by a wide margin. That is the opposite of the collection’s usual finding about long repeats, which are where the separable drafts and the long floats live — and it has the same cause seen from the other side. A pick in a long repeat is held at many crossings, so losing one matters less; a pick in a short repeat is held at few, so losing one can be all of them.

Which sharpens the essay’s own remark about the plain weave. A plain weave is at the most robust end of the balance condition and the least robust end of the repeat condition, and the two do not cancel: it survives structurally, because every pick is balanced, and its float trebles, because it had almost none. The shortest repeat is the one that cannot lose a thread quietly, whether or not it can lose one safely.

The same question asked of the weft #

Everything above deletes a column. Deleting a row is a different accident — a pick that was never laid, or one that broke and was withdrawn — and it is worth asking whether the asymmetry between warp and weft survives into the structure.

It does not. The matrix has no preferred direction, so deleting a row of a draft and deleting a column of its transpose are the same operation, and every count above applies to a missing pick as exactly as it does to a missing end. Forty-four per cent of the ways to lose a pick leave an end with nothing holding it; none of them splits the cloth; the float along the picks grows to three pick-lengths.

That is a small result and it is worth saying because the asymmetry in every other part of this subject is so strong. A warp fault condemns thirty times what a weft fault does; a warp fault runs the length of the piece and a weft fault its width; a warp is thousands of independent threads and a weft is one. All of those are facts about the loom and the piece.

The structure knows none of it. The draft is a matrix, the two directions are its two indices, and the criterion treats them alike — so the whole of the warp-and-weft asymmetry in this subject lives outside the matrix, in the machine that realises it. That is the clearest demonstration this collection has of where its own central object stops.

Where the model stops #

A broken end is not quite a deleted column. The neighbouring ends close up over the gap, so the cloth is locally denser on either side of the fault and the spacing across it is doubled rather than merely irregular. The matrix knows nothing of spacing, so the census reports the topology of the fault correctly and its geometry not at all.

A slack end is treated as a missing one. Structurally that is right — an end under no tension holds nothing down — but it is still physically there, taking up room and catching light, and the cloth’s appearance is not the appearance of a cloth with a gap.

The enumeration is at four by four, which is the repeat this collection can exhaust. The six-by-six control is a sample rather than an enumeration, so its 0.15 per cent is a rate rather than a count, and nothing here says how the rate behaves as the repeat grows.

And the criterion is still topological. It says a pick held by one crossing is attached, and it is right, and a pick held by one crossing in a slippery yarn will slide. Friction is the quantity it cannot see, and a fault is exactly where that blindness matters most: the picks that remain after an end is lost are held by fewer crossings, which is a question of degree that a yes-or-no criterion cannot report.

The generalisation #

Removing a constraint and reversing one are different operations with different failure modes, and the reversal is the dangerous one. A system that loses a component degrades; a system that has a component working backwards can partition. That distinction survives translation into anything with a directed relation in it — a dependency graph, a lattice of implications, a network of supports — and it is the reason the biggest failures are so often small and wrong rather than large and absent.

The second lesson is about classifications that were built for intact objects. The criterion here was designed to answer a yes-or-no question about a draft somebody meant. Asked about a damaged one it returns a third answer, which is not a refinement of the first two but a statement that the question does not apply. Any classification carried across from working systems to broken ones should be checked for that, and the check is cheap: run it on damage and see whether the categories still partition.

Who found it, and when #

The integrity criterion is this collection’s own and has been applied to intact drafts since it was built. Applying it to a damaged one appears to be new, and the enumeration certainly is: nothing in the literature counts what a broken end does to a weave, because the question requires the criterion and the criterion is recent.

That a broken end leaves floats and a loose pick is, of course, entirely familiar to weavers — it is what the fault looks like. What the census adds is how often, which is a number nobody had, and the zero, which is a statement about the operation rather than about weaving.

Where the ladder goes next #

The complementary perturbation is the smaller one: a pick made in the wrong shed, where a single row is replaced rather than a column removed — and where the cloth can fall into layers, which the missing end never manages.

Sideways, the fault’s cost was settled a rung ago — a broken end condemns thirty times what a mispick does — and the pairing of the two is the useful one: the fault that costs most is the one the structure survives, and the fault that costs least is the one that can break it.